Linear Algebra Quiz

Let u = (1, 2, 3) and v = (4, 5, 6). Compute 2u - 3v.

• (-10, -11, -12)

• (10, 11, 12)

• (-10, -11, 12)

• (10, 11, -12)

Determine if the vectors u = (1, 2) and v = (3, 4) are linearly independent.

• Yes

• No

• Only if they are orthogonal.

• Only if they are normalized.

Find the value of k such that the vectors u = (1, k) and v = (2, 3) are linearly dependent.

• k = 3/2

• k = 2/3

• k = 6

• k = 0

Given the vectors u = (1, 2, 3) and v = (4, 0, -1), compute their dot product u ⋅ v.

• 1

• 2

• 3

• 4

What is the length (magnitude) of the vector v = (3, 4)?

• 3

• 4

• 5

• 7

Normalize the vector u = (1, 1).

• (1/√2, 1/√2)

• (1, 1)

• (√2, √2)

• (0, 0)

Are the vectors u = (1, 2, 3), v = (4, 5, 6), and w = (7, 8, 9) linearly independent?

• Yes

• No

• It depends on the scalar multiples

• It is impossible to determine without further information

Find scalars a and b such that a(1, 2) + b(3, 1) = (0, 0) (assuming a and b are not both zero).

• a = 0, b = 0

• a = 1, b = -2

• a = 2, b = -1

• a = 3, b = -1

Determine if the set of vectors S = {(1, 0, 0), (0, 1, 0), (1, 1, 0)} spans R³.

• Yes

• No

• Only if the components are integers.

• Only if the components are positive.

What is the dimension of the vector space P₂ (polynomials of degree ≤ 2)?

• 1

• 2

• 3

• Infinite